Calculus Volume 2 · 免费预览
微分方程基础
微分方程把函数与其导数联系起来，其阶是所出现的最高阶导数，解是恒等满足该方程的函数，而初始条件则从一般解族中选出特定解。
1. What a differential equation is
What a differential equation is
2. Verifying and the general solution
Verifying and the general solution
3. Where differential equations come from
Where differential equations come from
4. A first table of solutions
A first table of solutions
5. Order classification
The equation y double prime plus y = 0 is classified as what order?
6. Linearity check
Which of the following differential equations is linear?
7. Verifying a solution
To verify that y = C e^(2x) solves y prime = 2y, what does one do?
8. Initial condition role
In the initial value problem y prime = ky with y(0) = 5, what does the condition contribute?
9. Existence and uniqueness
The existence and uniqueness theorem for y prime = f(x, y) with y(x_0) = y_0 guarantees what, when f is well-behaved?
10. General solution constant count
A second order differential equation's general solution contains how many arbitrary constants? Answer with a number.
11. Differential equation unknown
In a differential equation like y prime = ky, the unknown being solved for is what kind of mathematical object?
12. Logistic order
The logistic equation dP/dt = kP(1 - P/K) involves only the first derivative of P. What order is it? Answer with a number.
13. Particular solution check
For y prime = 3y with y(0) = 2, the particular solution is 2 e^(3x). At x = 0, what value does it take? Answer with a number.
14. Nonlinear example
The pendulum equation theta double prime plus sin theta = 0 is nonlinear because of which function applied to the unknown? Answer with the function name in lowercase.